Thread: Infinity or a Finite Neigbourhood of Infinity?

Should we treat infinity as: a Finite Neigbourhood of Infinity?

Such a question may be relevant to functions like:

f (t) = lim (x -> infinity) t*x*cos x + t^2*sin x

wich would be +/-t^2 for x large and x falling where cos x is zero.

Could you clarify what your thinking is?

Originally Posted by talanum1

Should we treat infinity as: a Finite Neigbourhood of Infinity?

Such a question may be relevant to functions like:

f (t) = lim (x -> infinity) t*x*cos x + t^2*sin x

wich would be +/-t^2 for x large and x falling where cos x is zero.

That is ridiculous.

It has no meaning and no conceivable meaning.

If infinity is a point it may fall on the real number k*pi/2 for k the infinite natural number, so that that function reduce to t^2 for this specific infinity. Since cos k*pi/2 is zero.

If infinity is really a finite neigborhood of infinity it doesn't.

You could have a problem with "specific infinity"?

Infinity is not a point in the sense you're using.

Then you must logicly explain the sense.

The real projective line does not do it.

Er, no. It's your idea, so you have to logically explain it. As best as I could understand your post, you're using infinity wrong.

There is no difficult issue to it. All that is needed is a function F(k) that maps Natural numbers to Real numbers. Defined as:

F (k) = pi*k/2

then we can use the deduced f (t) together with the specification F (infinity). It is just that f (t) = t^2 only exist like this if the mapping F (k) is so defined at infinity.

We therefore need a symbol to state this (like f(t) E* F(k)) and include two Real lines (for image) or a Real line and a Natural number lattice (for preimage) in the reasoning.

We then "just" show this does not lead to inconsistencies.

An "ideal point" needs only the definition of the notion of finite dimension (back generalised) and compositon of points being a line (continuum of points). Infinity falls somewhere in the continuum and gets treated specially but not in regard to the first line of this paragraph.